IQC analysis of linear constrained MPC

1. IQC analysis of linear constrained MPC W.P. Heath*, G. Li*, A.G. Wills†, B. Lennox* *University of Manchester †University of Newcastle, Australia

2. TLAs: • MPC: Model Predictive Control • IQC: Integral Quadratic Constraint Also: • KKT: Karush-Kuhn-Tucker • KYP: Kalman-Yakubovich-Popov • LMI: Linear Matrix Inequality

3. Overview • IQC theory • Familiar examples • Quadratic programming and sector bounds • Robustness of MPC • Example • Computation • Zames-Falb multipliers

4. Overview • IQC theory • Familiar examples • Quadratic programming and sector bounds • Robustness of MPC • Example • Computation • Zames-Falb multipliers

5. IQC theory:

6. IQC notation:

7. IQC theory:

8. Overview • IQC theory • Familiar examples • Quadratic programming and sector bounds • Robustness of MPC • Example • Computation • Zames-Falb multipliers

9. Example: small gain theorem

10. Example: multivariable circle criterion f

11. Overview • IQC theory • Familiar examples • Quadratic programming and sector bounds • Robustness of MPC • Example • Computation • Zames-Falb multipliers

12. Quadratic programmingand sector bounds

13. Quadratic programmingand sector bounds

14. MPC stability We can use IQC theory to test stability of many MPC structures. For example: Remark: there is no requirement for MPC internal model to match the plant

15. Overview • IQC theory • Familiar examples • Quadratic programming and sector bounds • Robustness of MPC • Example • Computation • Zames-Falb multipliers

16. Diagonal augmentation

17. So we can combine uncertainty and static nonlinearities: • D represents uncertainty • f represents static nonlinearity

18. MPC robust stability For MPC we can combine • the quadratic programming nonlinearity • the model uncertainty into a single block satisfying a single IQC. It remains to test the condition on the remaining linear element.

19. Overview • IQC theory • Familiar examples • Quadratic programming and sector bounds • Robustness of MPC • Example • Computation • Zames-Falb multipliers

20. Example

21. Example in standard form

22. Example: • 10 step horizon • 2x2 plant • IQC made up from four separate blocks (two nonlinearities and 2 uncertainties) • Weight on states is 1/k

23. Overview • IQC theory • Familiar examples • Quadratic programming and sector bounds • Robustness of MPC • Example • Computation • Zames-Falb multipliers

24. KYP lemma The stability condition is equivalent to an LMI • For MPC: • LMI equation dimension grows linearly with horizon • LMI solution dimension is independent of horizon

25. Overview • IQC theory • Familiar examples • Quadratic programming and sector bounds • Robustness of MPC • Example • Computation • Zames-Falb multipliers

26. Multipliers and IQCs • Multipliers allow more general choice of IQC • This in turn leads to less conservative stability results • Natural expression and generalisaiton of (for example): • Commutant sets for structured uncertainty • Nonlinear results such as Popov stability criterion

27. Zames-Falb multipliers Zames and Falb introduced general class of multipliers (1968) f is - bound - monotone nondecreasing - slope restricted Safanov and Kulkarni considered their application to multivariable nonlinearities (2000) independent of path

28. Zames-Falb multipliers for quadratic programming Result: Zames-Falb multipliers can be applied to the quadratic programme nonlinearity. Proof: via KKT conditions and convexity. Compare: - Fiacco et al: sensitivity analysis in nonlinear programming - Geometry of multiparametric quadratic programming

29. Conclusion • IQC theory provides a robust stability test of simple MPC loops (with arbitrary horizon) • We have illustrated the test for a 2x2 system and a 10 step horizon MPC • Current work: • How should we optimise multipliers? • How conservative is the test?